zenon/participatory-budgeting
1
0
Fork 0
Code Issues Pull Requests Projects Releases Wiki Activity

Finish presentation

Browse Source 
Add `\pause` for individual bullet points, add conclusion slide and
finish axioms.
This commit is contained in:
Tobias Eidelpes 2020-04-29 16:08:03 +02:00
parent 04c8f74578
commit 579fb445d3
1 changed files with 172 additions and 91 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

263
talk/talk.tex
View File

@ -6,6 +6,7 @@
\usecolortheme{dolphin} \usecolortheme{dolphin}
\usepackage{graphicx} \usepackage{graphicx}
\usepackage{caption}
\usepackage{tikz} \usepackage{tikz}
\usepackage{dsfont} \usepackage{dsfont}
@ -29,62 +30,77 @@
\section{Introduction} \section{Introduction}
\begin{frame} \begin{frame}
\frametitle{What is Participatory Budgeting?} \frametitle{What is Participatory Budgeting?} \pause
\begin{quote} \begin{quote}
Participatory Budgeting (PB) is a democratic process in which Participatory Budgeting (PB) is a democratic process in which
community members decide how to spend part of a public budget. community members decide how to spend part of a public budget.
\end{quote} \end{quote} \pause
\vspace{1cm} \vspace{1cm}
\begin{itemize} \begin{itemize}
\setlength{\itemsep}{1.1\baselineskip} \setlength{\itemsep}{1.1\baselineskip}
\item Participatory part: community members propose projects \item Participatory part: community members propose projects \pause
\item Budgeting part: each project requires a fixed amount of money \item Budgeting part: each project requires a fixed amount of money
\pause
\item Goal: Fund the \emph{best} projects without exceeding the budget \item Goal: Fund the \emph{best} projects without exceeding the budget
\end{itemize} \end{itemize}
\end{frame} \end{frame}
\begin{frame} \begin{frame}
\frametitle{How does it work?} \frametitle{A bit of background information} \pause
\begin{itemize}
\setlength{\itemsep}{1\baselineskip}
\item PB originated in Porto Alegre in 1990s \pause
\item Rapid expansion and influx of immigrants increased inequality
\pause
\item PB as a tool to combat inequality \pause
\item PB spread to Europe and North America \pause
\item Today \$300M allocated
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{How does it work?} \pause
\tikzstyle{blue} = [rectangle,rounded \tikzstyle{blue} = [rectangle,rounded
corners=3pt,draw=blue!50,fill=blue!20,node distance=1cm] corners=3pt,draw=blue!50,fill=blue!20]
\tikzstyle{red} = [rectangle,rounded corners=3pt,draw=red!50,fill=red!20] \tikzstyle{red} = [rectangle,rounded corners=3pt,draw=red!50,fill=red!20]
\tikzstyle{arrow} = [line width=1pt,->,>=triangle 60,draw=black!70] \tikzstyle{arrow} = [line width=1pt,->,>=triangle 60,draw=black!70]
\begin{center} \begin{center}
\begin{tikzpicture}[shorten >= 2pt,level distance=1.3cm] \begin{tikzpicture}[shorten >= 2pt,node distance=1.3cm]
\node[blue](design){Design the process} \node [blue] (design) {Design the process}; \pause
child { node [blue](collect){Collect ideas} \node [blue] (collect) [below of=design] {Collect ideas};
child { node [blue](develop){Develop feasible projects} \draw [arrow] (design) to (collect); \pause
child { node [red](vote){Vote on projects} \node [blue] (develop) [below of=collect] {Develop feasible
child { node [red](aggregate){Aggregate votes \& fund winners} projects};
} \draw [arrow] (collect) to (develop); \pause
} \node [red] (vote) [below of=develop] {Vote on projects};
} \draw [arrow] (develop) to (vote); \pause
} \node [red] (aggregate) [below of=vote] {Aggregate votes \& fund
; winners};
\draw [arrow] (design) to (collect);
\draw [arrow] (collect) to (develop);
\draw [arrow] (develop) to (vote);
\draw [arrow] (vote) to (aggregate); \draw [arrow] (vote) to (aggregate);
\onslide<1->
\end{tikzpicture} \end{tikzpicture}
\end{center} \end{center}
\end{frame} \end{frame}
\section{Modeling a PB scenario}
\begin{frame} \begin{frame}
\frametitle{A general framework for PB} \frametitle{A general framework for PB} \pause
\begin{itemize} \begin{itemize}
\setlength{\itemsep}{1\baselineskip} \setlength{\itemsep}{1\baselineskip}
\item Projects $P=\{p_1,\dots,p_m\}$ \item Projects $P=\{p_1,\dots,p_m\}$ \pause
\begin{itemize} \begin{itemize}
\setlength{\itemsep}{.7\baselineskip} \setlength{\itemsep}{.7\baselineskip}
\item Each project $p\in P$ has associated cost \item Each project $p\in P$ has associated cost \pause
$c(p):P\rightarrow\mathbb{R}$ $c(p):P\rightarrow\mathbb{R}$
\item Projects are either divisible or indivisible (discrete) \item Projects are either divisible or indivisible (discrete)
\pause
\end{itemize} \end{itemize}
\item Select a set $A\subseteq P$ as \emph{winning projects} not \item Select a set $A\subseteq P$ as \emph{winning projects} not
exceeding total budget $B$ exceeding total budget $B$ \pause
\begin{itemize} \begin{itemize}
\setlength{\itemsep}{.7\baselineskip} \setlength{\itemsep}{.7\baselineskip}
\item Discrete case: $\sum_{p\in A}c(p)\leq B$ \item Discrete case: $\sum_{p\in A}c(p)\leq B$ \pause
\item Divisible case: $\mu(p): P\rightarrow [0,1]$ with \item Divisible case: $\mu(p): P\rightarrow [0,1]$ with
$\sum_{p\in A}c(\mu(p))\leq B$ $\sum_{p\in A}c(\mu(p))\leq B$
\end{itemize} \end{itemize}
@ -95,51 +111,56 @@
\frametitle{A general framework for PB ctd.} \frametitle{A general framework for PB ctd.}
\begin{itemize} \begin{itemize}
\setlength{\itemsep}{1\baselineskip} \setlength{\itemsep}{1\baselineskip}
\item Voters $V=\{v_1,\dots,v_n\}$ \item Voters $V=\{v_1,\dots,v_n\}$ \pause
\begin{itemize} \begin{itemize}
\setlength{\itemsep}{.4\baselineskip} \setlength{\itemsep}{.4\baselineskip}
\item Express preferences over individual projects in $P$ or \item Express preferences over individual projects in $P$ or
over subsets in $\mathcal{P}(P) := \{P_v\,|\,P_v\subseteq P\}$ over subsets in $\mathcal{P}(P) := \{P_v\,|\,P_v\subseteq
P\}$ \pause
\item Preference elicitation is dependent on the input method \item Preference elicitation is dependent on the input method
(approval-based, ranked orders) (approval-based, ranked orders) \pause
\end{itemize} \end{itemize}
\item Aggregation methods \item Aggregation methods \pause
\begin{itemize} \begin{itemize}
\setlength{\itemsep}{.4\baselineskip}
\item Aggregation methods combine votes to determine a set \item Aggregation methods combine votes to determine a set
of winning projects of winning projects \pause
\item Are usually tied to the input method \item Are usually tied to the input method \pause
\item Rules are used to select projects w.r.t. desired \item Rules are used to select projects w.r.t. desired
properties of the outcome (fairness, welfare) properties of the outcome (fairness, welfare)
\end{itemize} \end{itemize}
\end{itemize} \end{itemize}
\end{frame} \end{frame}
\section{Algorithms for PB}
\begin{frame} \begin{frame}
\frametitle{Input and aggregation methods} \frametitle{Input and aggregation methods}
Example input methods: Example input methods: \pause
\begin{itemize} \begin{itemize}
\item Approval preferences \item Approval preferences \pause
\item Ranked orders \item Ranked orders \pause
\item Utility-based preferences \item Utility-based preferences \pause
\end{itemize} \end{itemize}
\vspace{0.2cm} \vspace{0.2cm}
Example aggregation methods: Example aggregation methods: \pause
\begin{itemize} \begin{itemize}
\item Maximizing social welfare \item Maximizing social welfare \pause
\item Greedy selection \item Greedy selection \pause
\item Fairness-based selection \item Fairness-based selection \pause
\end{itemize} \end{itemize}
\vspace{0.2cm} \vspace{0.2cm}
Aggregation methods depend on how voters elicit their preferences. Aggregation methods depend on how voters elicit their preferences.
\end{frame} \end{frame}
\begin{frame} \begin{frame}
\frametitle{Approval-based budgeting methods} \frametitle{Approval-based budgeting methods} \pause
\begin{itemize} \begin{itemize}
\item Voters approve a subset of projects \item Suitable for discrete PB \pause
\item Voter preferences are assumed to be \emph{dichotomous} \item Voters approve a subset of projects \pause
\item Voter preferences are assumed to be \emph{dichotomous} \pause
\item A \emph{satisfaction function} provides a metric for voter \item A \emph{satisfaction function} provides a metric for voter
satisfaction satisfaction \pause
\end{itemize} \end{itemize}
\begin{block}{An approval-based budgeting scenario} \begin{block}{An approval-based budgeting scenario}
A budgeting scenario is a tuple $E = (P,V,c,B)$ where $P = A budgeting scenario is a tuple $E = (P,V,c,B)$ where $P =
@ -148,7 +169,7 @@
$p\in P$ with its cost $c(p)$ and $B\in\mathbb{N}$ is a budget limit. A $p\in P$ with its cost $c(p)$ and $B\in\mathbb{N}$ is a budget limit. A
voter $v\in V$ specifies $P_v\subseteq P$, containing all approved voter $v\in V$ specifies $P_v\subseteq P$, containing all approved
items. items.
\end{block} \end{block} \pause
\begin{block}{Budgeting method $\mathcal{R}$} \begin{block}{Budgeting method $\mathcal{R}$}
A budgeting method $\mathcal{R}$ takes a budgeting scenario $E$ and A budgeting method $\mathcal{R}$ takes a budgeting scenario $E$ and
returns a bundle $A\subseteq P$ where the total cost of the items in returns a bundle $A\subseteq P$ where the total cost of the items in
@ -157,18 +178,18 @@
\end{frame} \end{frame}
\begin{frame} \begin{frame}
\frametitle{Satisfaction functions} \frametitle{Satisfaction functions} \pause
\begin{block}{Satisfaction function} \begin{block}{Satisfaction function}
A satisfaction function $sat : 2^P\times 2^P\rightarrow\mathbb{R}$ with A satisfaction function $sat : 2^P\times 2^P\rightarrow\mathbb{R}$ with
a set $P$ of items, a voter $v$ and her approval set $P_v$ and a bundle a set $P$ of items, a voter $v$ and her approval set $P_v$ and a bundle
$A\subseteq P$ provides the satisfaction $sat(P_v,A)$ of $v$ from the $A\subseteq P$ provides the satisfaction $sat(P_v,A)$ of $v$ from the
bundle $A$. The set of approved items by $v$ that end up in the winning bundle $A$. The set of approved items by $v$ that end up in the winning
bundle is denoted by $A_v = P_v\cap A$. bundle is denoted by $A_v = P_v\cap A$.
\end{block} \end{block} \pause
\begin{exampleblock}{$sat_\#(P_v,A)$} \begin{exampleblock}{$sat_\#(P_v,A)$}
$sat_\#(P_v,A) = |A_v|$: The satisfaction of voter $v$ is the number of $sat_\#(P_v,A) = |A_v|$: The satisfaction of voter $v$ is the number of
funded items that are approved. funded items that are approved.
\end{exampleblock} \end{exampleblock} \pause
\end{frame} \end{frame}
\begin{frame} \begin{frame}
@ -176,7 +197,7 @@
\begin{exampleblock}{$sat_\$(P_v,A)$} \begin{exampleblock}{$sat_\$(P_v,A)$}
$sat_{\$}(P_v,A) = \sum_{p\in A_v}{c(p) = c(A_v)}$: The satisfaction of voter $sat_{\$}(P_v,A) = \sum_{p\in A_v}{c(p) = c(A_v)}$: The satisfaction of voter
$v$ is the total cost of her approved and funded items. $v$ is the total cost of her approved and funded items.
\end{exampleblock} \end{exampleblock} \pause
\begin{exampleblock}{$sat_{0/1}(P_v,A)$} \begin{exampleblock}{$sat_{0/1}(P_v,A)$}
\[ sat_{0/1}(P_v,A) = \[ sat_{0/1}(P_v,A) =
\begin{cases} \begin{cases}
@ -190,17 +211,17 @@
\end{frame} \end{frame}
\begin{frame} \begin{frame}
\frametitle{Rules for selecting a winning bundle} \frametitle{Rules for selecting a winning bundle} \pause
{\Large Let $sat$ be a satisfaction function:} {\Large Let $sat$ be a satisfaction function:} \pause
\begin{block}{Max rules} \begin{block}{Max rules}
The rule $\mathcal{R}_{sat}^m$ selects a bundle which maximizes the sum The rule $\mathcal{R}_{sat}^m$ selects a bundle which maximizes the sum
of voters' satisfaction: $\mathsf{max}_{A\subseteq P}\sum_{v\in of voters' satisfaction: $\mathsf{max}_{A\subseteq P}\sum_{v\in
V}{sat(P_v,A)}$ V}{sat(P_v,A)}$
\end{block} \end{block} \pause
\begin{block}{Greedy rules} \begin{block}{Greedy rules}
The rule $\mathcal{R}_{sat}^g$ iteratively adds an item $p\in P$ to $A$, The rule $\mathcal{R}_{sat}^g$ iteratively adds an item $p\in P$ to $A$,
seeking to maximize $\sum_{v\in V}{sat(P_v,A\cup\{p\})}$. seeking to maximize $\sum_{v\in V}{sat(P_v,A\cup\{p\})}$.
\end{block} \end{block} \pause
\begin{block}{Proportional greedy rules} \begin{block}{Proportional greedy rules}
The rule $\mathcal{R}_{sat}^p$ iteratively adds an item $p\in P$ to $A$ The rule $\mathcal{R}_{sat}^p$ iteratively adds an item $p\in P$ to $A$
seeking to maximize the sum of satisfaction per unit of cost. seeking to maximize the sum of satisfaction per unit of cost.
@ -208,49 +229,51 @@
\end{frame} \end{frame}
\begin{frame} \begin{frame}
\frametitle{Example budgeting scenarios} \frametitle{Example budgeting scenarios} \pause
\begin{block}{A budgeting scenario} \begin{block}{A budgeting scenario}
Items $P = \{p_2,p_3,p_4,p_5,p_6\}$ and their associated cost $p_i$ Items $P = \{p_2,p_3,p_4,p_5,p_6\}$ and their associated cost $p_i$
where $p_i$ costs $i$. Budget limit $B=10$ and 5 voters vote where $p_i$ costs $i$. Budget limit $B=10$ and 5 voters vote
$v_1=\{p_2,p_5,p_6\}$, $v_2=\{p_2,p_3,p_4,p_5\}$, $v_3=\{p_3,p_4,p_5\}$, $v_1=\{p_2,p_5,p_6\}$, $v_2=\{p_2,p_3,p_4,p_5\}$, $v_3=\{p_3,p_4,p_5\}$,
$v_4=\{p_4,p_5\}$ and $v_5=\{p_6\}$. $v_4=\{p_4,p_5\}$ and $v_5=\{p_6\}$.
\end{block} \end{block} \pause
\begin{exampleblock}{Combining max rule with $sat_\#$} \begin{exampleblock}{Combining max rule with $sat_\#$}
Under $\mathcal{R}_{|A_v|}^m$ the winning bundle is $\{p_2,p_3,p_5\}$. Under $\mathcal{R}_{|A_v|}^m$ the winning bundle is $\{p_2,p_3,p_5\}$.
The total satisfaction is 8. The total satisfaction is 8.
\end{exampleblock} \end{exampleblock} \pause
\begin{exampleblock}{Combining greedy rule with $sat_\#$} \begin{exampleblock}{Combining greedy rule with $sat_\#$}
Under $\mathcal{R}_{|A_v|}^g$ the winning bundle is $\{p_4,p_5\}$ (first Under $\mathcal{R}_{|A_v|}^g$ the winning bundle is $\{p_4,p_5\}$ (first
selecting $p_5$). The total satisfaction is 7. selecting $p_5$). The total satisfaction is 7.
\end{exampleblock} \end{exampleblock} \pause
\begin{exampleblock}{Combining max rule with $sat_{0/1}$} \begin{exampleblock}{Combining max rule with $sat_{0/1}$}
Under $\mathcal{R}^m_{sat_{0/1}}$ the winning bundle is $\{p_4,p_6\}$, Under $\mathcal{R}^m_{sat_{0/1}}$ the winning bundle is $\{p_4,p_6\}$,
achieving max satisfaction. achieving max satisfaction.
\end{exampleblock} \end{exampleblock}
\end{frame} \end{frame}
\section{Complexity of PB algorithms}
\begin{frame} \begin{frame}
\frametitle{Complexity of budgeting algorithms} \frametitle{Complexity of algorithms} \pause
\begin{itemize} \begin{itemize}
\setlength{\itemsep}{1\baselineskip} \setlength{\itemsep}{1\baselineskip}
\item Computing winners using greedy rules ($\mathcal{R}^g_{sat}$) can \item Computing winners using greedy rules ($\mathcal{R}^g_{sat}$) can
be done in polynomial time: be done in polynomial time: \pause
\begin{itemize} \begin{itemize}
\setlength{\itemsep}{.4\baselineskip} \setlength{\itemsep}{.4\baselineskip}
\item these rules are defined through efficient iterative \item these rules are defined through efficient iterative
processes processes \pause
\item however: making a series of locally optimal choices does \item however: making a series of locally optimal choices does
not always lead to a globally optimal choice not always lead to a globally optimal choice \pause
\item $\mathcal{R}^g_{|A_v|}$ is similar to $k$-Approval and \item $\mathcal{R}^g_{|A_v|}$ is similar to $k$-Approval and
knapsack voting knapsack voting \pause
\end{itemize} \end{itemize}
\item Max rules ($\mathcal{R}^m_{sat}$) are generally NP-hard \item Max rules ($\mathcal{R}^m_{sat}$) are generally NP-hard \pause
\begin{itemize} \begin{itemize}
\setlength{\itemsep}{.4\baselineskip} \setlength{\itemsep}{.4\baselineskip}
\item $\mathcal{R}^m_{|A_v|}$ can be solved in polynomial time \item $\mathcal{R}^m_{|A_v|}$ can be solved in polynomial time
because one dimension is fixed because one dimension is fixed \pause
\item $\mathcal{R}^m_{sat_{0/1}}$: finding a bundle with at least a \item $\mathcal{R}^m_{sat_{0/1}}$: finding a bundle with at least a
given total satisfaction is NP-hard given total satisfaction is NP-hard \pause
\item satisfaction functions can be modeled as integer linear \item satisfaction functions can be modeled as integer linear
programs programs
\end{itemize} \end{itemize}
@ -258,66 +281,118 @@
\end{frame} \end{frame}
\begin{frame} \begin{frame}
\frametitle{Complexity of budgeting algorithms ctd.} \frametitle{Complexity of algorithms ctd.}
{\large Dealing with \emph{intractability}:} {\large Dealing with \emph{intractability}:} \pause
\vspace{.3cm} \vspace{.3cm}
\begin{itemize} \begin{itemize}
\setlength{\itemsep}{1\baselineskip} \setlength{\itemsep}{1\baselineskip}
\item Provide an approximation algorithm, sacrificing exactness \item Provide an approximation algorithm, sacrificing exactness \pause
\begin{itemize} \begin{itemize}
\setlength{\itemsep}{0.4\baselineskip} \setlength{\itemsep}{0.4\baselineskip}
\item No algorithm with approx. ratio better than $1-1/\epsilon$ \item No algorithm with approx. ratio better than $1-1/\epsilon$
exists for $\mathcal{R}^m_{sat_{0/1}}$ exists for $\mathcal{R}^m_{sat_{0/1}}$ \pause
\end{itemize} \end{itemize}
\item Fixed-parameter tractability: fix one parameter to solve problem \item Fixed-parameter tractability: fix one parameter to solve problem
in reasonable amount of time in reasonable amount of time \pause
\begin{itemize} \begin{itemize}
\setlength{\itemsep}{0.4\baselineskip} \setlength{\itemsep}{0.4\baselineskip}
\item Fix parameter $m$ (the number of items) \item Fix parameter $m$ (the number of items) \pause
\item Fix parameter $n$ (the number of voters) \item Fix parameter $n$ (the number of voters)
\end{itemize} \end{itemize}
\end{itemize} \end{itemize}
\end{frame} \end{frame}
\section{Axioms for PB algorithms}
\begin{frame} \begin{frame}
\frametitle{Comparing budgeting algorithms} \frametitle{Comparing algorithms} \pause
By defining desirable axioms, different budgeting algorithms can \begin{itemize}
be compared: \setlength{\itemsep}{.9\baselineskip}
\begin{block}{Discount Monotonicity} \item Compare algorithms by using axioms \pause
Suppose a budgeting algorithm $\mathcal{R}$ outputs a winning set of \item Axioms provide \emph{guidelines} for choosing an algorithm \pause
projects $A$. The cost of project $p\in A$ is lowered (discounted) \item Some might be desirable, others are not a drawback, if not
compared to the previous cost. $\mathcal{R}$ should output another satisfied \pause
winning set $A'$ where project $p$ is not implemented to a lesser \item Desirable properties: \emph{fairness}, \emph{strategyproofness}
degree. \end{itemize}
\end{block}
\end{frame} \end{frame}
\begin{frame} \begin{frame}
\frametitle{Axiom Examples} \frametitle{Comparing algorithms—Discount Monotonicity} \pause
\begin{block}{Discount Monotonicity}
Suppose a budgeting algorithm $\mathcal{R}$ outputs a winning set of
projects $A$. The cost of project $p\in A$ is lowered (discounted)
compared to the previous cost. $\mathcal{R}$ should output another
winning set $A'$ where project $p$ is not implemented to a lesser
degree.
\end{block} \pause
\begin{block}{A budgeting scenario} \begin{block}{A budgeting scenario}
Items $P = \{p_2,p_3,p_4,p_5,p_6\}$ and their associated cost $p_i$ Items $P = \{p_2,p_3,p_4,p_5,p_6\}$ and their associated cost $p_i$
where $p_i$ costs $i$. Budget limit $B=10$ and 5 voters vote where $p_i$ costs $i$. Budget limit $B=10$ and 5 voters vote
$v_1=\{p_2,p_5,p_6\}$, $v_2=\{p_2,p_3,p_4,p_5\}$, $v_3=\{p_3,p_4,p_5\}$, $v_1=\{p_2,p_5,p_6\}$, $v_2=\{p_2,p_3,p_4,p_5\}$, $v_3=\{p_3,p_4,p_5\}$,
$v_4=\{p_4,p_5\}$ and $v_5=\{p_6\}$. $v_4=\{p_4,p_5\}$ and $v_5=\{p_6\}$.
\end{block} \end{block} \pause
\begin{exampleblock}{Discount Monotonicity Example} \begin{exampleblock}{Discount Monotonicity Example}
Under $\mathcal{R}^m_{|A_v|}$ the winning bundle is $\{p_2,p_3,p_5\}$. Under $\mathcal{R}^m_{|A_v|}$ the winning bundle is $\{p_2,p_3,p_5\}$.
After discounting $p_2$ to $p_1$, we still get $\{p_1,p_3,p_5\}$. The After discounting $p_2$ to $p_1$, we still get $\{p_1,p_3,p_5\}$.
total cost is one unit less but the total satisfaction remains the same.
\end{exampleblock} \end{exampleblock}
\end{frame} \end{frame}
\section{Future Directions} \begin{frame}
\frametitle{Comparing algorithms—Limit Monotonicity} \pause
\begin{block}{Limit Monotonicity}
A budgeting method $\mathcal{R}$ satisfies Limit Monotonicity if for a
pair of budgeting scenarios $E=(P,V,c,B)$, $E'=(P,V,c,B+1)$ and with no
project that costs exactly $B+1$, for each project $p\in P$
$p\in\mathcal{R}(E)\implies p\in\mathcal{R}(E')$ holds.
\end{block} \pause
\begin{block}{A budgeting scenario}
Items $P=\{a_1,b_1,c_1\}$ (all with unit cost), budget limit $B=1$ and 4
voters vote $v_1=\{a_1\}$, $v_2=\{a_1,b_1\}$, $v_3=\{b_1,c_1\}$ and
$v_4=\{c_1\}$.
\end{block} \pause
\begin{exampleblock}{Limit Monotonicity Example}
Under $\mathcal{R}^m_{sat_{0/1}}$ a winning bundle might be $\{b_1\}$.
Increasing the budget limit to 2 results in $\{a_1,c_1\}$.
\end{exampleblock}
\end{frame}
\begin{frame} \begin{frame}
\frametitle{Future Areas of Interest} \frametitle{Applying the two axioms to the example algorithms} \pause
\begin{itemize}
\setlength{\itemsep}{1\baselineskip}
\item Discount Monotonicity: \pause
\begin{itemize}
\setlength{\itemsep}{.4\baselineskip}
\item All 3 algorithms ($\mathcal{R}^m_{|A_v|}$,
$\mathcal{R}^g_{|A_v|}$, $\mathcal{R}^m_{sat_{0/1}}$)
satisfy discount monotonicity \pause
\item Decreasing the cost increases the attractiveness of an
item \pause
\item Not true for algorithms that measure satisfaction by
maximizing the cost of winning projects \pause
\end{itemize}
\item Limit Monotonicity: \pause
\begin{itemize}
\setlength{\itemsep}{.4\baselineskip}
\item No discussed algorithm satisfies limit monotonicity \pause
\item Greedy heuristics fail when a project fits into the new
budget, providing higher satisfaction \pause
\item Increasing the budget might lead to projects being dropped
\end{itemize}
\end{itemize}
\end{frame}
\section{Conclusion}
\begin{frame}
\frametitle{Conclusion} \pause
\begin{itemize} \begin{itemize}
\item Multi-dimensional constraints \setlength{\itemsep}{1\baselineskip}
\item Hybrid models \item PB: What is it? \pause
\item Complex resident preferences \item Computational aspects \pause
\item Market-based approaches \item Aggregation algorithms \pause
\item The role of information \item Complexity of algorithms \pause
\item Research spanning the entire PB process \item Comparison of algorithms using axioms
\end{itemize} \end{itemize}
\end{frame} \end{frame}
@ -326,6 +401,12 @@
\Large \Large
Thank you for your attention! \\ Thank you for your attention! \\
Questions \& Answers Questions \& Answers
\begin{figure}
\centering
\includegraphics[width=.5\textwidth]{voting_referendum.png}
\captionsetup{labelformat=empty}
\caption{\tiny [\url{https://xkcd.com/2225}]}
\end{figure}
\end{frame} \end{frame}
\end{document} \end{document}